Definition

then x∈Ax \in A. Such an AA may be called a ≺\prec-inductive subset of SS. The relation ≺\prec is well-founded if the only ≺\prec-inductive subset of SS is SS itself.

Note that this is precisely what is necessary to validate induction over ≺\prec: if we can show that a statement is true of x∈Sx\in S whenever it is true of everything ≺\prec-below xx, then it must be true of everything in SS. In the presence of excluded middle it is equivalent to other commonly stated definitions; see Formulations in classical logic below.

Formulations in classical logic

While the definition above follows how a well-founded relation is generally used (namely, to prove properties of elements of SS by induction), it is complicated. Two alternative formulations are given by the following:

The relation ≺\prec has no infinite descent (usually attributed to Pierre de Fermat) if there exists no sequence⋯≺x2≺x1≺x0\cdots \prec x_2 \prec x_1 \prec x_0 in SS. (Such a sequence is called an infinite descending sequence.)

The relation ≺\prec is classically well-founded if every inhabited subset AA of SS has a member x∈Ax \in A such that no t∈At \in A satisfies t≺xt \prec x. (Such an xx is called a minimal element of AA.)

In classical mathematics, both of these conditions are equivalent to being well-founded. In constructive mathematics, we may prove that a well-founded relation has no infinite descent (see Proposition 1), but not the converse, and that a classically well-founded relation is well-founded (see Proposition 2), but not the converse.

Remark

We note that classical well-foundedness is really too strong for constructive (i.e., intuitionistic) mathematics: if there exists an inhabited relation that is classically well-founded, then excluded middle follows. (This holds true in any topos; for a proof, see here.) On the other hand, the infinite descent condition is too weak to be of much use in constructive mathematics. It is the inductive notion of well-foundedness that is just right.

Note however that in predicative mathematics, the definition of well-founded may be impossible to even state, and so either of these alternative definitions would be preferable, provided classical logic is used.

Even in constructive predicative mathematics, (1) is strong enough to establish the Burali-Forti paradox (when applied to linear orders). In material set theory, (2) is traditionally used to state the axiom of foundation, although the impredicative definition could also be used as an axiom scheme (and must be in constructive versions). In any case, either (1) or (2) is usually preferred by classical mathematicians as simpler.

To round out the discussion we prove the following two results, both valid in intuitionistic mathematics:

Proposition

If (X,≺)(X, \prec) is a well-founded relation and A⊆XA \subseteq X has no minimal element, then AA is empty.

This result makes it trivial to infer (under classical logic) that classical well-foundedness is a consequence of well-foundedness. It also shows that well-foundedness rules out infinite descent (intuitionistically), since an infinite descent sequence has no minimal element.

So, suppose zz is an element such that y∈Uy \in U whenever y≺zy \prec z. We must show z∈Uz \in U. Claim: z∉Az \notin A. For if z∈Az \in A, then zz would be a minimal element of AA (as y≺z⇒y∈U⇒y∉Ay \prec z \Rightarrow y \in U \Rightarrow y \notin A). But this negates the assumption that AA has no minimal element.

Proposition

Proof

Let ≺\prec be a classically well-founded relation on XX, and let UU be an inductive subset. We must show that every element x∈Xx \in X belongs to UU. Since UU is inductive, it suffices to show that every u≺xu \prec x belongs to UU, i.e. we may assume given a uu such that u≺xu\prec x and show u∈Uu\in U. But under this assumption we have that ≺\prec is inhabited, so according to Remark 1, the law of excluded middle follows and we might as well then argue classically. The argument is well-known but we include it for completeness: under classical well-foundedness, if an inductive subset UU is not the entirety of XX, then the complement ¬U\neg U has a minimal element yy. In that case, v≺yv \prec y implies v∈¬¬U=Uv \in \neg\neg U = U, but then y∈Uy \in U since UU is inductive, contradiction. Hence U=XU = X and in particular u∈Uu \in U, which is what we wanted.

To bring us full circle: in classical set theory we may prove that if (X,≺)(X, \prec) has no infinite descent, then ≺\prec is classically well-founded. For suppose an inhabited subset P⊆XP \subseteq X (say with an element x∈Px \in P) failed to have a least element. Then we can find an infinite descent sequence xn∈Px_n \in P with x0=xx_0 = x, by choosing at each stage xn+1∈Px_{n+1} \in P such that xn+1≺xnx_{n+1} \prec x_n. Technically this requires the use of dependent choice, but generally this is felt to be a mild choice principle (that is true even for intuitionistic mathematics).

Coalgebraic formulation

Many inductive or recursive notions may also be packaged in coalgebraic terms. For the concept of well-founded relation, first observe that a binary relation ≺\prec on a set XX is the same as a coalgebra structure θ:X→P(X)\theta\colon X \to P(X) for the covariant power-set endofunctor on SetSet, where y≺xy \prec x if and only if y∈θ(x)y \in \theta(x).

In this language, a subseti:U↪Xi\colon U \hookrightarrow X is ≺\prec-inductive, or θ\theta-inductive, if in the pullback

Then, as usual, the PP-coalgebra (X,θ)(X, \theta) is well-founded if every θ\theta-inductive subset U↪XU \hookrightarrow X is all of XX.

Other relevant notions may also be packaged; for example, the PP-coalgebra XX is extensional if θ:X→PX\theta\colon X \to P X is monic. See also well-founded coalgebra.

Simulations

Given two sets SS and TT, each equipped with a well-founded relation ≺\prec, a functionf:S→Tf\colon S \to T is a simulation of SS in TT if 1. f(x)≺f(y)f(x) \prec f(y) whenever x≺yx \prec y and 2. given t≺f(x)t \prec f(x), there exists y≺xy \prec x with t=f(y)t = f(y).

In coalgebraic language, a simulation S→TS \to T is simply a PP-coalgebra homomorphism f:S→Tf\colon S \to T. Condition (1), that ff is merely ≺\prec-preserving, translates to the condition that ff is a colax morphism of coalgebras, in the sense that there is an inclusion

Properties

Every well-founded relation is irreflexive; that is, x⊀xx \nprec x. Sometimes one wants a reflexive version ⪯\preceq of a well-founded relation; let x⪯yx \preceq y if and only x≺yx \prec y or x=yx = y. Then the requirement that xx be a minimal element of a subset AA states that t⪯xt \preceq x only if t=xt = x. But infinite descent or direct proof by induction still require ≺\prec rather than ⪯\preceq.

The axiom of foundation in material set theory states precisely that the membership relation ∈\in on the proper class of all pure sets is well-founded. In structural set theory, accordingly, one uses well-founded relations in building structural models of well-founded pure sets.

Examples

Let SS be the set of natural numbers, and let x≺yx \prec y if yy is the successor of xx: y=x+1y = x + 1. That this relation is well-founded is the usual principle of mathematical induction.

Again let SS be the set of natural numbers, but now let x≺yx \prec y if x<yx \lt y in the usual order. That this relation is well-founded is the principle of strong induction.

More generally, let SS be a set of ordinal numbers (or even the proper class of all ordinal numbers), and let x≺yx \prec y if x<yx \lt y in the usual order. That this relation is well-founded is the principle of transfinite induction.

Similarly, let SS be a set of pure sets (or even the proper class of all pure sets), and let x≺yx \prec y if x∈yx \in y. That this relation is well-founded is the axiom of foundation.

Let SS be the set of integers, and let x≺yx \prec y mean that xx properly divides yy: y/xy/x is an integer other than ±1\pm{1}. This relation is also well-founded, so one can prove properties of integers by induction on their proper divisors.